nLab alternating representation

Definition

For $S_n$ a symmetric group, its alternating representation is the 1-dimensional linear representation

$$alt \;\; S_n \longrightarrow GL(1)$$

which sends even permutations to $+1$ and odd permutations to $-1$.

More generally, for $G$ any finite group and $H \subset G$ a subgroup of index 2, then the corresponding alternating representation of $G$ sends elements of $H$ to $+1$ and all other elements to -1.

This reduces to the previous special case by setting $G \coloneqq S_n$ and $H \coloneqq A_n \subset S_n$ the alternating group.

Related concepts

• trivial representation

• regular representation

• permutation representation

References

For instance

• João Pedro Martins dos Santos, Def. 3 in Representation Theory of Symmetric Groups, 2012 (pdf)